arXiv subjects

Xiaotao Sun

Publications and source records attributed to Xiaotao Sun.

At least 19 recordsLinked to original sources

arXiv2025

The étale fundamental group and $F$-divided sheaves in characteristic $p>0$

We investigate how the étale fundamental group controls local systems in characteristic $p$, namely $F$-divided sheaves. In analogy with Grothendieck-Malcev's results for discrete groups, we show that if a morphism $f \colon Y \to X$ of smooth projective varieties over $k=\bar{k}$ induces a surjection on the étale fundamental groups, then the pullback functor ${\rm Fdiv}(X)\to {\rm Fdiv}(Y)$ is fully faithful. If $f$ is surjective and the induced map is an isomorphism, then the functor is an equivalence. These results extend the theorem of Esnault-Mehta on the triviality of $F$-divided sheaves over simply connected varieties.

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arXiv2023

Frobenius splitting of moduli spaces of parabolic bundles

Let $C$ be a nonsingular projective curve over an algebraically closed field of characteristic $p>0$ and $I\subset C$ be a finite set. If $\mathcal{U}_{C,\,ω}$ denotes the moduli space of semistable parabolic bundles of rank $r$ and degree $d$ on $C$ with parabolic structures determined by $ω=(k,\{\vec n(x),\vec a(x)\}_{x\in I})$, we prove that $\mathcal{U}_{C,\,ω}$ is \textit{$F$-split} for generic $C$ and generic choice of $I$ when $p>3r$.

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arXiv2020

A finite dimensional proof of Verlinde Formula

We prove two recurrence relations among dimensions $$D_g(r,d,ω):={\rm dim}\,{\rm H}^0(\mathcal{U}_{C,\,ω},Θ_{\mathcal{U}_{C,\,ω}})$$ of spaces of generalized theta functions on moduli spaces $\mathcal{U}_{C,\,ω}$. By using of these recurrence relations, an explicit formula (Verlinde formula) of $D_g(r,d,ω)$ is proved (See Theorem 4.3).

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arXiv2020

Globally F-regular type of Moduli spaces

We prove moduli spaces of semistable parabolic bundles and generalized parabolic sheaves with fixed determinant on a smooth projective curve are globally $F$-regular type.

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arXiv2019

Slope inequalities and a Miyaoka-Yau type inequality

For a minimal smooth projective surface $S$ of general type over a field of characteristic $p>0$, we prove that $K^2_S\le 32χ(\cal{O}_S).$ Moreover, if $18χ(\cal{O}_S) 0$, which answers completely a question of Shepherd-Barron.

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arXiv2019

Surfaces on the Severi line in positive characteristics

Let $X$ be a minimal surface of general type over an algebraically closed field $\mathbf{k}$ of $\mathrm{char}.(\mathbf{k})=p\ge 0$. If the Albanese morphism $a_X:X\to \mathrm{Alb}_X$ is generically finite onto its image, we formulate a constant $c(X,L)\ge 0$ for a very ample line bundle $L$ on $\mathrm{Alb}_X$ such that $c(X,L)=0$ if and only if $\dim \mathrm{Alb}_X=2$ and $a_X: X\to \mathrm{Alb}_X$ is a double cover. A refined Severi inequality $$K^2_X\ge (4+{\rm min}\{\,c(X,L),\,\frac{1}{3}\,\})χ(\mathcal{O}_X)$$ is proved. Then we prove that $K^2_X=4χ(\mathcal{O}_X)$ if and only if the canonical model of $X$ is a flat double cover of an Abelian surface.

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arXiv2018

Globally F-regular type of moduli spaces and Verlinde formula

We prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves (GPS) with a fixed determinant on a smooth projective curve are globally F-regular type. As an application, we prove vanishing theorems on the moduli spaces of semistable parabolic sheaves on a singular curve, which combining with Factorization theorems in [24] and [25] give two recurrence relations among dimensions of spaces of generalized theta functions. By using of these recurrence relations, we prove an explicit formula (Verlinde formula) for the dimension of spaces of generalized theta functions.

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arXiv2017

Factorization of generalized Theta functions revisited

This survey is based on my lectures given in last a few years. As a reference, constructions of moduli spaces of parabolic sheaves and generalized parabolic sheaves are provided. By a refinement of the proof of vanishing theorem, we show, without using vanishing theorem, a new observation that ${\rm dim}\,H^0(\sU_C,Θ_{\sU_C})$ is independent of all of the choices for any smooth curves. The estimate of various codimension and computation of canonical line bundle of moduli space of generalized parabolic sheaves on a reducible curve are provided in Section 6, which is completely new.

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arXiv2017

Stratified bundles and Representation spaces

For a given stratified bundle $E$ on $X$, we construct an irreducible closed subvariety $\sN(E)_S$ of the so called representation space $R(\sO_{X_S},ξ_S,P)\to S$ such that $\sN(E)_S(\overline{\mathbb{F}}_q)$ contains a dense set of $(V,β)$ where $V$ is induced by a representation of $π_1^{{\rm \acute{e}t}}(X)$ (Theorem \ref{thm3.7}). As an application, we give a simply proof of the main theorem of \cite{EM} and its relative version (Theorem \ref{thm4.2}).

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arXiv2016

Remarks on Xiao's approach of Slope inequalities

We prove the slope inequality for a relative minimal surface fibration in positive characteristic via Xiao's approach. We also prove a better low bound for the slope of non-hyperelliptic fibrations.

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arXiv2011

Stratified bundles and étale fundamental group (new version)

This submission replaces the arXiv:1012.5381 submission with the same title, which had been withdrawn as it contained a mistake, repaired in this submission: on $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that all irreducible stratified bundles have rank 1 if and only if the commutator $[π_1, π_1]$ of the étale fundamental group $π_1$ is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $ π_1 $ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.

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arXiv2011

Stratified bundles and étale fundamental group

v2: A few typos corrected, a few formulations improved. On $X$ projective smooth over an algebraically closed field of characteristic $p>0$, we show that irreducible stratified bundles have rank 1 if and only if the commutator $[π_1^{{\rm \acute{e}t}}, π_1^{{\rm \acute{e}t}}]$ of the étale fundamental group is a pro-$p$-group, and we show that the category of stratified bundles is semi-simple with irreducible objects of rank 1 if and only if $ π_1^{{\rm \acute{e}t}}$ is abelian without $p$-power quotient. This answers positively a conjecture by Gieseker.

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arXiv2010

Elliptic curves in moduli space of stable bundles

Let $M$ be the moduli space of rank $2$ stable bundles with fixed determinant of degree $1$ on a smooth projective curve $C$ of genus $g\ge 2$. When $C$ is generic, we show that any elliptic curve on $M$ has degree (respect to anti-canonical divisor $-K_M$) at least 6, and we give a complete classification for elliptic curves of degree $6$. Moreover, if $g>4$, we show that any elliptic curve passing through the generic point of $M$ has degree at least $12$. We also formulate a conjecture for higher rank.

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arXiv2009

Stability of sheaves of locally closed and exact forms

For any smooth projective variety $X$ of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$ with $μ(Ω^1_X)>0$. If ${\rm T}^{\ell}(Ω^1_X)$ ($0<\ell 0$ and $Ω^1_X$ is semi-stable, the sheaf $B^2_X$ of exact 2-forms is also stable. Moreover, under the same condition, the sheaf $Z^1_X$ of closed 1-forms is stable when $p>3$, and $Z^1_X$ is semi-stable when $p=3$.

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arXiv2009

Frobenius morphism and semi-stable bundles

This article is the expanded version of a talk given at the conference: Algebraic geometry in East Asia 2008, Seoul. In this notes, I intend to give a brief survey of results on the behavior of semi-stable bundles under the Frobenius pullback and direct images. Some results are new.

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arXiv2008

Remarks on lines and minimal rational curves

We determine all of lines in the moduli space $M$ of stable bundles for arbitrary rank and degree. A further application of minimal rational curves is also given in last section.

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arXiv2008

Direct images of bundles under Frobenius morphisms

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. For any vector bundle $W$ on $X$, we prove that instability of $F_*W$ is bounded by instability of $W\otimes{\rm T}^{\ell}(Ω^1_X)$ ($0\le \ell\le n(p-1)$)(Corollary \ref{cor3.8}). When $X$ is a smooth projective curve of genus $g\ge 2$, it implies $F_*W$ being stable whenever $W$ is stable.

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arXiv2006

Stability of direct images under Frobenius morphism

Let $X$ be a smooth projective variety over an algebraically field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. When ${\rm dim}(X)=1$, we prove that $F_*W$ is a stable bundle for any stable bundle $W$ (Theorem \ref{thm1.3}). As a step to study the question for higher dimensional $X$, we generalize the canonical filtration (defined by Joshi-Ramanan-Xia-Yu for curves) to higher dimensional $X$ (Theorem \ref{thm2.6}).

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